MATH 1610

Final - Practice 1 | General
1. Find the derivative of $y = x\sqrt{1 - x^2}$.

A) $\frac{-x}{\sqrt{1 - x^2}}$
B) $\frac{x}{\sqrt{1 - x^2}}$
C) $\sqrt{1 - x^2} - \frac{x^2}{\sqrt{1 - x^2}}$
D) $\sqrt{1 - x^2} + \frac{x^2}{\sqrt{1 - x^2}}$
E) $\sqrt{1 - x^2} + \frac{x^2}{2\sqrt{1 - x^2}}$

2. Evaluate $\lim_{x \to -4} \frac{x^2 + x - 12}{x^2 + 7x + 12}$.

A) DNE
B) $5$
C) $6$
D) $7$
E) $8$
F) $9$
G) $\frac{0}{0}$

3. Evaluate $\int \sin x - \csc^2 x \; dx$

A) $-\cos x - \sec x + C$
B) $-\cos x + \cot x + C$
C) $\cos x - \cot x + C$
D) $-\cos x + \csc x + C$
E) $\cos x - \csc x + C$

4. Evaluate $\frac{d}{dx} \int_{2x}^{3x} \frac{1}{\ln t} dt$

A) $\frac{3}{\ln 3x} - \frac{2}{\ln 2x}$
B) $\frac{3x}{\ln 3x} - \frac{2x}{\ln 2x}$
C) $\frac{3x}{\ln t} - \frac{2x}{\ln t}$
D) $\frac{1}{3x} - \frac{1}{2x}$
E) $\frac{1}{\ln 3x} - \frac{1}{\ln 2x}$

5. Find the deivative of $x^2y^2 + x^3 + y = 8$ at $(2, 0)$.

A) $0$
B) $-3$
C) $-6$
D) $-9$
E) $-12$
F) $-15$

6. $f(x) = e^{2x}. Evaluate $\lim_{h \to 0} \frac{f(0 + h) - f(0)}{h}$.

A) $\frac{0}{0}$
B) DNE
C) $0$
D) $1$
E) $2$
F) $3$

7. Find the $x$ value of the point on the curve $y = 3x + 2$ closest to the point $(1, 0)$.

A) $-\frac{3}{2}$
B) $-1$
C) $-\frac{1}{2}$
D) $0$
E) $\frac{1}{2}$
F) $1$
G) $\frac{3}{2}$

8. An icicle in the shape of a cone is growing in volume at the rate of $1 \; \text{cm}^3/\text{min}$. The height always equals twice the radius of the base. When the height equals $10$ cm, how fast is the height increasing? (Hint: $V = \frac{1}{3} \pi r^2 h$.)

A) $\frac{1}{25\pi}$
B) $\frac{1}{75\pi}$
C) $\frac{1}{225\pi}$
D) $\frac{1}{315\pi}$
E) $\frac{1}{400\pi}$

9. Let $y = xe^x$. Where is $y$ increasing?

A) $x > 0$
B) $x > -1$
C) $x > 2$
D) $x < -1$
E) $x < 0$

10. Find the derivative of $y = \cot^{-2}(x)$.

A) $2 \cot^{-3}(x) \csc^2(x)$
B) $-2 \cot^{-3}(x) \csc^2(x)$
C) $2\cot^{-2}(x) \csc(x)$
D) $-2\cot^{-2}(x) \csc(x)$
E) $\-csc^{-4}(x)$

11. Solve $f'(x) = 3x^2 + e^x$ for $f(x)$ when $f(0) = 2$.

A) $f(x) = x^3 + e^x + 1$
B) $f(x) = x^3 + e^x + 2$
C) $f(x) = 6x + e^x$
D) $f(x) = x^3 + e^x$
E) $f(x) = 3x^2 + e^x + 2$

12. Evaluate $\int_1^2 3x^2 - 2x \; dx$

A) $4$
B) $3$
C) $2$
D) $1$
E) $0$
F) $-1$

13. Find the general antiderivative of $3x^{1/2} + x^{-1/2}$.

A) $\3x^{3/2} - x^{1/2} + C$
B) $2x^{3/2} + 2x^{1/2} + C$
C) $2x^{3/2} - 2x^{1/2} + C$
D) $\frac{3}{2} x^{-1/2} + \frac{1}{2}x^{-3/2} + C$
E) $\frac{9}{2} x^{3/2} - \frac{1}{2}x^{1/2} + C$

14. Find the horizontal asymptote for $y = \frac{1 + 7x^2 + 3x^3}{x^4 - x}$.

A) $y = -1$
B) $y = 0$
C) $y = 1$
D) $y = 2$
E) $y = 3$

15. Find the vertical asymptote(s) for $y = \frac{1 + 7x^2 + 3x^3}{x^4 - x}$.

A) $x = 0$
B) $x = 0, x = 1$
C) $x = -1, x = 0, x = 1$
D) $x = -1, x = 1$
E) $x = 1$

16. Find the absolute maximum of the function $f(x) = x^4 - 8x^2$ on the interval $-1 \le x \le 3$.

A) $-20$
B) $-16$
C) $-7$
D) $0$
E) $3$
F) $9$
G) $12$

17. Find the absolute minimum of the function $f(x) = x^4 - 8x^2$ on the interval $-1 \le x \le 3$.

A) $-20$
B) $-16$
C) $-7$
D) $0$
E) $3$
F) $9$
G) $12$

18. $f(x) = x^4 - 6x^2$ and $f'(x) = 4x^3 - 12x$. Where is $f(x)$ concave up?

A) $x > 1$
B) $x < -1, x > 1$
C) $-1 < x < 1$
D) $-\sqrt{3} < x < 0, x > \sqrt{3}$
E) $x < -\sqrt{3}, 0 < x < \sqrt{3}$

19. Evaluate $\lim_{x \to -4} \frac{x}{(4 + x)^6}$.

A) $\frac{1}{0}$
B) $-\infty$
C) $\infty$
D) DNE
E) None of the above

20. Evaluate $\int \frac{2x^3}{\sqrt{x^4 + 9}} dx$

A) $x^4\sqrt{x^4 + 9} + C$
B) $\frac{1}{2} \sqrt{x^4 + 9} + C$
C) $\frac{1}{\sqrt{x^4 + 9}} + C$
D) $\frac{6x^2\sqrt{x^4 + 9} - 4x^6(x^4 + 9)^{-1/2}}{x^4 + 9} + C$
E) $\sqrt{x^4 + 9} + C$

21. $y = \frac{x^3 + 3x}{x}$. Find $y''$.

A) $0$
B) $1$
C) $2$
D) $3$
E) $4$
F) $5$
G) $6$

22. Evaluate $\lim_{x \to 1} \frac{\ln x - x + 1}{(x - 1)^2}$

A) $-1/6$
B) $-1/4$
C) $-1/2$
D) $1/2$
E) $1/4$
F) $1/6$

23. Find the derivative of $y = \arctan(x^2)$.

A) $\frac{2x}{1 + x^2}$
B) $\frac{2x}{1 + x^4}$
C) $-\tan^{-2}(x^2)\sec^2(x^2)2x$
D) $\frac{1}{1 + x^4}$
E) $\tan^{-2}(x^2)\sec^2(x^2)2x$

24. Evaluate $\int_0^{1/2} \frac{1}{\sqrt{1 - x^2}} dx$

A) $0$
B) $\frac{\pi}{6}$
C) $\frac{\pi}{4}$
D) $\frac{\pi}{3}$
E) $\frac{\pi}{2}$

25. Find the derivative of $y = \frac{\ln^2 x}{x}$.

A) $\frac{2 \ln x - \ln^2 x}{x^2}$
B) $\frac{2x \ln x - \ln^2 x}{x^2}$
C) $\frac{2 \ln x}{x}$
D) $\frac{2 - 2 \ln x}{x^2}$
E) $\frac{\frac{1}{x} - \ln^2 x}{x^2}$

26. Evaluate $\int \frac{9x^4 + 5x^2}{x^{1/2}} dx$

A) $\frac{\frac{9x^5}{5} + \frac{5x^3}{3}}{\frac{2x^{3/2}}{3}} + C$
B) $2x^{9/2} + 2x^{5/2} + C$
C) $9x^{7/2} + 5x^{3/2} + C$
D) $\frac{x^{1/2} (36x^3 + 10x) - (9x^4 + 5x^2) \frac{1}{2}x^{1/2}}{x}$
E) $\frac{81}{2}x^{9/2} + \frac{25}{2} x^{5/2} + C$

27. Evaluate $\int \frac{3x^2}{\sqrt{1 - x^6}} dx$

A) $2 \sqrt{1 - x^6} + C$
B) $3x^2 \arcsin (x^3) + C$
C) $\arcsin(x^3) + C$
D) $\frac{6x\sqrt{1 - x^6} + 9x^7(1 - x^6)^{-1/2}}{1 - x^6}$
E) $\ln | \sqrt{1 - x^6} | + C$

28. Evaluate $\lim_{x \to 1} \frac{\frac{1}{3 - x} - \frac{1}{x + 1}}{x - 1}$.

A) DNE
B) $-\frac{1}{4}$
C) $\frac{1}{4}$
D) $-\frac{1}{3}$
E) $\frac{1}{3}$
F) $-\frac{1}{2}$
G) $\frac{1}{2}$

29. Evaluate $\lim_{x \to 3} \frac{3 - \sqrt{12 - x}}{3 - x}$

A) $\frac{0}{0}$
B) DNE
C) $\frac{1}{6}$
D) $0$
E) $-\frac{1}{6}$
F) $\frac{1}{2}$
G) $-\frac{1}{2}$

30. Which of the following graphs most closely has the following properties:
  • Increasing $-1 < x < 0, 0 < x < 1, x > 1$. Decreasing $x < -1$.
  • Concave down $x < -1, -1 < x < 0, x > 1$. Concave up $0 < x < 1$.
  • Vertical asymptotes $x = -1, x = 1$. Horizontal Asymptote $y = 1$.
  • $f(0) = 0$

A) !!q30-1.png!!
B) !!q30-2.png!!
C) !!q30-3.png!!
D) !!q30-4.png!!
E) !!q30-5.png!!